From a physical point of view a fluid is a material that lacks shear strength, i.e. that deforms, or “flows”, when subject to shear stress. The strain rate for a given shear stress is of course highly variable, and defines the viscosity of the fluid. From a chemical thermodynamic point of view it is convenient to distinguish between different types of fluids. There are fluids with relatively low densities and low viscosities, which tend to be highly mobile in planetary environments. These are often referred to as “volatiles”, and are typically composed of species in the system C–O–H–N–S–F–Cl, with inert gases (particularly He) also important in gas giants. There are also fluids with generally higher densities than volatiles, which also have much higher viscosities, typically by several orders of magnitude. If such fluids exist at equilibrium with solids of broadly similar bulk composition we call them melts (Chapter 10). Melts in terrestrial planets are chiefly silicates, although natural carbonate melts also exist, as do metallic melts in planetary cores. Melts in icy satellites, in contrast, are likely to be composed chiefly of species in the system C–O–H–N. A third type of fluids are liquids at conditions that are far removed from equilibrium with solids of similar bulk composition, but that may contain species in solution that crystallize their own solids.

Introduction

As discussed in the previous chapters (see, in particular, Section 6.3), if an isotropic field is Gaussian, its dependence structure is completely identified by the angular correlation function and its harmonic transform, that is, the angular power spectrum. For non-Gaussian fields, the dependence structure becomes much richer, and higher order correlation functions are of interest. In turn, this leads to the analysis of so-called higher order angular power spectra, which we investigated in Chapter 6. Cumulant angular power spectra are identically zero for Gaussian fields, and hence they also provide natural tools to test for non-Gaussianity: this is a topic of the greatest importance in modern cosmological data analysis (see [51, 60]). Indeed, the validation of the Gaussian assumption is urged by the necessity to provide firm grounds to statistical inference on cosmological parameters, which is dominated by likelihood approaches. More importantly, tests for Gaussianity are needed to discriminate among competing scenarios for the physics of the primordial epochs: here, the currently favoured inflationary models predict (very close to) Gaussian CMB fluctuations, whereas other models yield different observational consequences (see [8, 18, 19, 20, 67, 184]). Tests for non-Gaussianity are also powerful tools to detect systematic effects in the outcome of the experiments. For these reasons, very many papers have focussed on testing for non-Gaussianity on CMB, some of them by means of topological properties of Gaussian fields, some others through spherical wavelets, or by harmonic space methods, see for instance [18, 35, 36, 37, 117, 119, 130, 164, 165, 175, 176, 198], and the references therein.

The First Law of Thermodynamics, like all conservation laws, is expressed mathematically by an identity relationship. As such, it is incapable of predicting the direction in which a natural process will occur. For example, on the basis of energy conservation alone it is not possible to decide whether heat flows from a hot body to a colder one, or the other way around. We know that heat flows down a temperature gradient, but this does not follow from the First Law. Similarly, energy conservation cannot predict that ice will melt at 20°C, or that water will freeze at -20°C, and it cannot predict that a gas will expand to fill all of the volume available to it.

Another law of nature is required to predict the direction of spontaneous changes. By “spontaneous” we mean a process that occurs in nature in the direction towards equilibrium and without outside intervention. For example, heat flow down a temperature gradient is a spontaneous process. It is possible to transfer heat from a cold body to a hotter one, but this requires “outside intervention” in the form of a heat pump, which uses mechanical energy to accomplish a process that is not “naturally spontaneous”. As soon as the expenditure of mechanical energy ceases the spontaneous process takes over and the cold body heats up at the expense of the hotter one.

In the summer of 1935 young C. F. S. Sharpe undertook an excellent adventure. Having acquired a car, he drove 16 000 miles through 28 American states and 3 Canadian provinces. His quest was unusual: He was out to demonstrate that mass movement of debris over the Earth’s surface is an important geological process. After he returned he wrote up his observations in a book (Sharpe, 1938) that forms the basis of our modern understanding of gravity-driven mass motions for the evolution of the Earth’s landscape. The current era of space exploration has greatly broadened the reach of the processes he described: Mass motion is important on bodies ranging from tiny asteroids and comets only a few kilometers in diameter up to the largest moons and planets. Its action has been observed on every solid body in the Solar System.

Several people have worked with the authors in the last years to develop the material covered in this monograph. In particular, following the order of the book, we wish to mention Ivan Nourdin and David Nualart for recent developments on the generalized method of moments for Gaussian subordinated processes, and relationships with Stein's method; Jean-Renaud Pycke for spectral representations of isotropic random fields; Paolo Baldi for the characterizations of spherical harmonic coefficients under isotropy; Paolo Baldi, Gerard Kerkyacharian and Dominique Picard for the stochastic analysis of standard needlets, and Xiaohong Lan for the needlets bispectrum; Daryl Geller (who introduced Mexican needlets with Azita Mayeli) for the extension of the needlet paradigm to random sections of spin fiber bundles, We learned a lot from discussions with Mauro Piccioni and lgor Wigman, who have also provided very useful comments on an earlier draft, as did PhD students Mirko D'Ovidio and Claudio Durastanti.

The material of this book is strongly motivated by Cosmological applications, and it has benefited enormously from a decade-long interaction of the first author with physicists providing insights. Suggestions, and applications to real data: we mention in particular (in alphabetic order) Amedeo Balbi, Paolo Cabella, Giancarlo de Gasperis, Frode Hansen, Michele Liguori, Sabino Matarrese, Paolo Natoli, Davide Pietrobon, Gianluca Polenta, Oystein Rudjord, Sandro Scodeller and Nicola Vittorio. Frode Hansen is to be thanked also for some insightful comments on the CMB description parts.

The liquid state extends from the melting point to the boiling point. Beyond the critical point fluids with liquid-like densities transition continuously to fluids with gas-like densities. A liquid close to its freezing temperature may differ significantly in such properties as viscosity, microscopic structure and chemical behavior from a liquid of the same composition near its boiling or critical points. For this reason it is convenient to define a melt as a liquid that is at, or very near, its freezing point. A melt is therefore saturated, or nearly so, in a solid phase (or assemblage) of broadly similar bulk composition. The exact meaning of “broadly similar” will remain undefined, but will become clear from the context of this and the following chapter, in which we will discuss electrolyte solutions. There is a parallel between this definition of melt and that of vapor, which is a gas that is at equilibrium with its liquid.

This chapter focuses on the ways in which melts form in planetary interiors. Because several excellent and up-to-date textbooks on igneous petrology are available (see, Winter, 2001; McBirney, 2006; Philpotts & Ague, 2009), and the research literature in the field is vibrant, I will not discuss processes of magma evolution and crystallization. There is no point in repeating here what is explained in much greater detail elsewhere. It is important to recall that a magma is an assemblage of melt, suspended solids and dissolved volatiles.